Volume 26, Issue 2
The Sensitivity of the Exponential of an Essentially Nonnegative Matrix
DOI:

J. Comp. Math., 26 (2008), pp. 250-258

Published online: 2008-04

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• Abstract

This paper performs perturbation analysis for the exponential of an essentially nonnegative matrix which is perturbed in the way that each entry has a small relative perturbation. For a general essentially nonnegative matrix, we obtain an upper bound for the relative error in $2$-norm, which is sharper than the existing perturbation results. For a triangular essentially nonnegative matrix, we obtain an upper bound for the relative error in entrywise sense. This bound indicates that, if the spectral radius of an essentially nonnegative matrix is not large, then small entrywise relative perturbations cause small relative error in each entry of its exponential. Finally, we apply our perturbation results to the sensitivity analysis of RC networks and complementary distribution functions of phase-type distributions.

• Keywords

Essentially nonnegative matrix Matrix exponential Entrywise perturbation theory RC network Phase-type distribution

65F35 15A42.

@Article{JCM-26-250, author = {}, title = {The Sensitivity of the Exponential of an Essentially Nonnegative Matrix}, journal = {Journal of Computational Mathematics}, year = {2008}, volume = {26}, number = {2}, pages = {250--258}, abstract = { This paper performs perturbation analysis for the exponential of an essentially nonnegative matrix which is perturbed in the way that each entry has a small relative perturbation. For a general essentially nonnegative matrix, we obtain an upper bound for the relative error in $2$-norm, which is sharper than the existing perturbation results. For a triangular essentially nonnegative matrix, we obtain an upper bound for the relative error in entrywise sense. This bound indicates that, if the spectral radius of an essentially nonnegative matrix is not large, then small entrywise relative perturbations cause small relative error in each entry of its exponential. Finally, we apply our perturbation results to the sensitivity analysis of RC networks and complementary distribution functions of phase-type distributions.}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8622.html} }
TY - JOUR T1 - The Sensitivity of the Exponential of an Essentially Nonnegative Matrix JO - Journal of Computational Mathematics VL - 2 SP - 250 EP - 258 PY - 2008 DA - 2008/04 SN - 26 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8622.html KW - Essentially nonnegative matrix KW - Matrix exponential KW - Entrywise perturbation theory KW - RC network KW - Phase-type distribution AB - This paper performs perturbation analysis for the exponential of an essentially nonnegative matrix which is perturbed in the way that each entry has a small relative perturbation. For a general essentially nonnegative matrix, we obtain an upper bound for the relative error in $2$-norm, which is sharper than the existing perturbation results. For a triangular essentially nonnegative matrix, we obtain an upper bound for the relative error in entrywise sense. This bound indicates that, if the spectral radius of an essentially nonnegative matrix is not large, then small entrywise relative perturbations cause small relative error in each entry of its exponential. Finally, we apply our perturbation results to the sensitivity analysis of RC networks and complementary distribution functions of phase-type distributions.